Geodesic
Construction of Banach spaces using a generalization of the scalar product
Matematičeskie zametki, Tome 8 (1970) no. 1, pp. 67-76.

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A vector is assigned to every pair of elements of a linear space. If this vector is in $l_2$ and the relation between the element pair and the vector satisfies a certain system of axioms, then we call the space a Banach space. For such a space we introduce, as in the case of the usual scalar product, a series of concepts, and prove, as an example a theorem concerning orthogonal expansion. 
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     author = {N. N. Chaus},
     title = {Construction of {Banach} spaces using a generalization of the scalar product},
     journal = {Matemati\v{c}eskie zametki},
     pages = {67--76},
     publisher = {mathdoc},
     volume = {8},
     number = {1},
     year = {1970},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1970_8_1_a7/}
}
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N. N. Chaus. Construction of Banach spaces using a generalization of the scalar product. Matematičeskie zametki, Tome 8 (1970) no. 1, pp. 67-76. http://geodesic.mathdoc.fr/item/MZM_1970_8_1_a7/